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Unformatted text preview: e classes approach a limit? Example 1.5. Brand preference. Suppose there are three types of laundry detergent, 1, 2, and 3, and let Xn be the brand chosen on the nth purchase. Customers who try these brands are satisﬁed and choose the same thing again with probabilities 0.8, 0.6, and 0.4 respectively. When they change they pick one of the other two brands at random. The transition probability is 1 2 3 1 .8 .2 .3 2 .1 .6 .3 3 .1 .2 .4 Q. Do the market shares of the three product stabilize? 4 CHAPTER 1. MARKOV CHAINS Example 1.6. Inventory chain. We will consider the consequences of using an s, S inventory control policy. That is, when the stock on hand at the end of the day falls to s or below, we order enough to bring it back up to S . For simplicity, we suppose happens at the beginning of the next day. Let Xn be the amount of stock on hand at the end of day n and Dn+1 be the demand on day n + 1. Introducing notation for the positive part of a real number, ( x if x &gt; 0 + x = max{x, 0} = 0 if x 0 then we can write the chain in general as ( (Xn Dn+1 )+ Xn+1 = (S Dn+1 )+ if Xn &gt; s if Xn s In words, if Xn &gt; s we order nothing and begin the day with Xn units. If the demand Dn+1 Xn we end the day with Xn+1 = Xn Dn+1 . If the demand...
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